
We now aim to identify what the trivial motions of a D&B framework are, and to determine conditions for a framework to be infinitesimally rigid based on properties of \(\mathcal {R}_\mathcal {G}\) . Before we proceed, we note that the D&B rigidity function in \(\mathsf {SE}(3)\)  can be seen as a superposition of the rigidity functions associated with the classic distance rigidity theory [1]} and the \(\mathsf {SE}(3)\)  bearing rigidity theory [2]}. In particular, we note that \(\mathcal {R}_{\mathcal {G},d}: \mathbb {R}^{3N} \rightarrow \mathbb {R}^{|\mathcal {E}_u|\times 3N}\nabla _p \gamma _{d}\)  is the well-studied (distance) rigidity matrix, while \(\mathcal {R}_{\mathcal {G},b}:\mathsf {SE}^{3N}\rightarrow \mathbb {R}^{3|\mathcal {E}|\times 6N} \nabla _{(p,R)}\gamma _{\mathcal {G},b}\)  is the \(\mathsf {SE}(3)\)  bearing rigidity matrix. Note that \(\mathcal {R}_{\mathcal {G},d}\)  is associated with the framework \((\mathcal {G},p_\mathcal {G})\) , which is the projection of \((\mathcal {G},p_\mathcal {G},R_\mathcal {G})\)  to \(\mathbb {R}^3\) . With an appropriate permutation, \(P_R\) , of the columns of \(\mathcal {R}_{\mathcal {G}}\) , we have that
\( \widetilde{\mathcal {R}}_\mathcal {G} &\mathcal {R}_\mathcal {G} P_R \\&\hspace{-17.07182pt}= \begin{bmatrix}\frac{\partial \gamma _{1,d}}{\partial p_1} & \dots & \frac{\partial \gamma _{1,d}}{\partial p_N} & \frac{\partial \gamma _{1,d}}{\partial R_1} & \dots & \frac{\partial \gamma _{1,d}}{\partial R_N} \\ \vdots & & \ddots & & \vdots \\\frac{\partial \gamma _{M_\mathcal {G},d}}{\partial p_1} & \dots & \frac{\partial \gamma _{M_\mathcal {G},d}}{\partial p_N} & \frac{\partial \gamma _{M_\mathcal {G},d}}{\partial R_1} & \dots & \frac{\partial \gamma _{M_\mathcal {G},d}}{\partial R_N} \\\frac{\partial \gamma _{1,b}}{\partial p_1} & \dots & \frac{\partial \gamma _{1,b}}{\partial p_N} & \frac{\partial \gamma _{1,b}}{\partial R_1} & \dots & \frac{\partial \gamma _{1,b}}{\partial R_N} \\\vdots & & \ddots & & \vdots \\\frac{\partial \gamma _{M_\mathcal {G},b}}{\partial p_1} & \dots & \frac{\partial \gamma _{M_\mathcal {G},b}}{\partial p_N} & \frac{\partial \gamma _{M_\mathcal {G},b}}{\partial R_1} & \dots & \frac{\partial \gamma _{M_\mathcal {G},b}}{\partial R_N}\end{bmatrix},\) 
